SeriesParallel Circuits

1
Chapter 6

• Series-Parallel Circuits

2
Objectives

• Identify series-parallel relationships
• Analyze series-parallel circuits
• Analyze loaded voltage dividers
• Determine the loading effect of a voltmeter on a
  circuit
• Analyze a Wheatstone bridge circuit

3
Objectives

• Apply Thevenins theorem to simplify a circuit
  for analysis
• Apply the maximum power transfer theorem
• Apply the superposition theorem to circuit
  analysis

4
Identifying Series-Parallel Relationships

• A series-parallel circuit consists of
  combinations of both series and parallel current
  paths

5
Analysis of Series-Parallel Circuits

• Determine total resistance
• Determine all currents
• Determine all voltage drops

6
Total Resistance

• Identify the parallel resistances, and calculate
  the equivalent resistance(s)
• Identify the series resistance, and calculate the
  total resistance for the circuit

7
Total Current

• Using the total resistance and the source
  voltage, find the total current by applying Ohms
  law
• IT VS/RT

8
Branch Currents

• Using the current-divider formula, Kirchhoffs
  current law (KCL), Ohms law, or combinations of
  these, you can find the current in any branch of
  a series-parallel circuit

9
Unloaded Voltage Dividers

• A voltage divider produces an output which
  depends upon the values of the dividing resistors
• This voltage is the unloaded output voltage

10
Loaded Voltage Dividers

• When a load resistor RL is connected from the
  output to ground, the output voltage is reduced
  by an amount that depends on the value of RL

11
Load Current and Bleeder Current

• Bleeder current is the current left (I3) after
  the total load current is subtracted from the
  total current into the circuit

12
Loading Effect of a Voltmeter

• When measuring across a resistor, a voltmeter is
  connected in parallel with the resistor
• Being in parallel, the internal resistance of the
  voltmeter will have a loading effect on the
  circuit that is being measured
• Modern digital voltmeters (DMM) have an internal
  resistance of 10M?

13
Loading Effect of a Voltmeter

• If the meter resistance is at least ten times
  greater than the resistance across which it is
  connected, the loading effect can be neglected
• measurement error is less than 10

14
Wheatstone Bridge

• A Wheatstone bridge is used to precisely measure
  resistance
• A Wheatstone bridge is also applied with
  transducer measurements, to measure physical
  quantities such as temperature, strain, and
  pressure, where small transducer resistance
  changes may need to be precisely measured
• Tiny changes in transducer resistance will
  unbalance the bridge, thereby providing a
  measurement reading

15
Balanced Wheatstone Bridge

• The Wheatstone bridge is in the balanced bridge
  condition when the output voltage between
  terminals A and B is equal to zero

16
Unbalanced Wheatstone Bridge

• The unbalanced bridge, when VOUT is not equal to
  zero, is used to measure some transducer
  quantities, such as strain, temperature, or
  pressure
• The bridge is balanced at a known point, then the
  amount of deviation, as indicated by the output
  voltage, indicates the amount of change in the
  parameter being measured

17
Thevenins Theorem

• Thevenins theorem provides a method for
  simplifying a circuit to a standard equivalent
  form
• The Thevenin equivalent voltage (VTH) is the open
  circuit (no-load) voltage between two terminals
  in a circuit
• The Thevenin equivalent resistance (RTH) is the
  total resistance appearing between two terminals
  in a given circuit with all sources replaced by
  their internal resistances

18
Thevenin Equivalent of a Circuit
19
Summary of Thevenins Theorem

• Open the two terminals (remove any load) between
  which you want to find the Thevenin equivalent
  circuit
• Determine the voltage (VTH) across the two open
  terminals
• Determine the resistance (RTH) between the two
  open terminals with all sources replaced with
  their internal resistances (short voltage sources
  and open current sources)

20
Summary of Thevenins Theorem

• Connect VTH and RTH in series to produce the
  complete Thevenin equivalent for the original
  circuit
• Place the load resistor removed in Step 1 across
  the terminals of the Thevenin equivalent circuit.
  The load current and load voltage can now be
  calculated using only Ohms law. They have the
  same value as the load current and load voltage
  in the original circuit

21
Maximum Power Transfer

• Maximum power is transferred from a source to a
  load when the load resistance is equal to the
  internal source resistance

22
Maximum Power Transfer

• The source resistance, RS, of a circuit is the
  equivalent resistance as viewed from the output
  terminals using Thevenins theorem
• A typical application of the maximum power
  transfer theorem is in audio systems, where the
  speaker resistance must be matched to the audio
  power amplifier in order to obtain maximum output

23
Superposition Theorem

• Some circuits require more than one voltage or
  current source
• The superposition theorem is a way to determine
  currents and voltages in a circuit that has
  multiple sources by considering one source at a
  time

24
General statement of Superposition Theorem

• The current in any given branch of a
  multiple-source circuit can be found by
  determining the currents in that particular
  branch produced by each source acting alone, with
  all other sources replaced by their internal
  resistances. The total current in the branch is
  the algebraic sum of the individual source
  currents in that branch

25
Applying Superposition Theorem

• Take one voltage (or current) source at a time
  and replace the remaining voltage sources with
  shorts (and remaining current sources with opens)
• Determine the particular current or voltage that
  you want, just as if there were only one source
  in the circuit

26
Applying Superposition Theorem

• Take the next source in the circuit and repeat
  Steps 1 and 2 for each source
• To find the actual current in a given branch,
  algebraically sum the currents due to each
  individual source. Once the current is found,
  voltage can be determined by Ohms law

27
Summary

• A series-parallel circuit is a combination of
  both series paths and parallel paths
• To determine total resistance in a
  series-parallel circuit, identify the series and
  parallel relationships, and then apply the
  formulas for series resistance and parallel
  resistance
• To find the total current, apply Ohms law and
  divide the total voltage by the total resistance

28
Summary

• To determine branch currents, apply the
  current-divider formula, KCL, or Ohms law
• To determine voltage drops across any portion of
  a series-parallel circuit, use the
  voltage-divider formula, KVL, or Ohms law
• When a load resistor is connected across a
  voltage-divider output, the output voltage
  decreases

29
Summary

• A load resistor should be large compared to the
  resistance across which it is connected, in order
  that the loading effect may be minimized
• A balanced Wheatstone bridge can be used to
  measure an unknown resistance
• A bridge is balanced when the output voltage is
  zero. The balanced condition produces zero
  current through a load connected across the
  output terminals of the bridge

30
Summary

• An unbalanced Wheatstone bridge can be used to
  measure physical quantities using transducers
• Any two-terminal resistive circuit, no matter how
  complex, can be replaced by its Thevenin
  equivalent, made up of an equivalent resistance
  (RTH) in series with an equivalent voltage source
  (VTH)
• The maximum power transfer theorem states that
  the maximum power is transferred from a source to
  a load when Load Resistance equals Source
  Resistance